Alternating Direction Implicit Technique Research Articles

Polarization manipulation on step-index composite polymer beam splitters for photonics circuitry

This paper presents composite beam splitters realized with polymer materials for developing photonic integrated circuits. We used organic-inorganic hybrid polymer materials to form this composite beam splitter realized with step-index (SI) core profiles. We used the alternating direction implicit technique of the Rsoft CAD BeamPROP solver to design and analyze these beam splitters. We successfully examined and manipulated the beam splitter’s polarization dependency to obtain a 99% output efficiency with a 50:50 splitting ratio. The SI beam splitter exhibits an excess loss of 0.014 dB. When we apply polarized light in this beam splitter, the excess loss increases to 2 dB, and this loss gradually decreases as the angle of incident light increases. The excess loss reduces to 0.05 dB at the 31-degree angles of the incident polarized light. We also investigated the crosstalk of this beam splitter by varying the wavelength, and it is evident that the lowest crosstalk is −19.77 dB at the polarized angle of 31°.

Linearized Crank–Nicolson Scheme for the Two-Dimensional Nonlinear Riesz Space-Fractional Convection–Diffusion Equation

In this paper, we study the nonlinear Riesz space-fractional convection–diffusion equation over a finite domain in two dimensions with a reaction term. The Crank–Nicolson difference method for the temporal and the weighted–shifted Grünwald–Letnikov difference method for the spatial discretization are proposed to achieve a second-order convergence in time and space. The D’Yakonov alternating–direction implicit technique, which is effective in two–dimensional problems, is applied to find the solution alternatively and reduce the computational cost. The unconditional stability and convergence analyses are proved theoretically. Numerical experiments with their known exact solutions are conducted to illustrate our theoretical investigation. The numerical results perfectly confirm the effectiveness and computational accuracy of the proposed method.

A new stable finite difference scheme and its error analysis for two‐dimensional singularly perturbed convection–diffusion equations

AbstractThis work focuses on the numerical solution of two‐dimensional singularly perturbed convection–diffusion equations via a new stable finite difference (NSFD) scheme on a tensor product of two piecewise‐uniform Shishkin meshes. First, we convert the two‐dimensional equation into two one‐dimensional equations using the alternating direction implicit technique. A NSFD scheme has been developed using Taylor's series and the one‐dimensional equations. Here the truncation of Taylor's series is different from the classical finite difference scheme. The convergence analysis is also studied on a tensor product of two piecewise‐uniform Shishkin meshes. Numerical simulations confirm the theory. Moreover, from the numerical illustrations, it is also observed that the method is parameter uniform convergent.

Numerical simulation of three‐sided lid‐driven square cavity

AbstractIn this article, an incompressible, two‐dimensional (2D), time‐dependent Newtonian fluid flow in a square cavity is simulated using finite difference method and alternating direction implicit technique. Navier‐Stokes equations are solved numerically in stream function‐vorticity formulation. In order to verify the numerical solver, the one‐sided lid‐driven cavity is studied and the results are compared with relevant data in the literature. They were in a very good agreement. Furthermore, two distinguished unexplored cases of the three‐sided lid‐driven cavity have been investigated. In case (1), the upper and lower walls are translated to the right, while the left wall is translated upward and the right wall remains stationary. Moreover, in case (2), the upper wall is translated to the right but the lower wall is translated to the left, while the left wall is translated downward and the right wall remains stationary. However, stream function and vorticity values in addition to the location of primary and secondary vortices' centers inside the square cavity have been revealed at low and intermediate Reynolds numbers (Re), typically (Re = 100 to 2000). Moreover, the higher the Re, the more main primary vortex approaches the cavity center and the bigger the secondary vortices.

Uniformly convergent novel finite difference methods for singularly perturbed reaction–diffusion equations

AbstractIn this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction.

A high-order split-step finite difference method for the system of the space fractional CNLS

In this paper, the schemes based on the high-order quasi-compact split-step finite difference methods are derived for the one- and two-dimensional coupled fractional Schrodinger equations. In order to improve the computing efficiency, we adopt the split-step method for handling the nonlinearity. By using a high-order quasi-compact scheme in space, the numerical method improves the accuracy effectively. We prove the conservation laws, prior boundedness and unconditional error estimates of the quasi-compact finite difference scheme for the linear problem. Moreover, for the nonlinear problem, we show that the quasi-compact split-step finite difference method can also keep the conservation law in the mass sense. For solving the multi-dimensional problem, we combine the quasi-compact split-step method with the alternating direction implicit technique. At last, numerical examples are performed to illustrate our theoretical results and show the efficiency of the proposed schemes.

A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation

A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three‐dimensional fractional evolution equation. In our method, the Riemann‐Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one‐dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.

Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.

Interaction Between Two Mutually Orthogonal Heat-Generating Baffles in a Square Cavity

ABSTRACTLaminar free convection induced by two mutually orthogonal discrete heat-generating baffles in a two-dimensional square cavity is analyzed numerically. The computations were carried out for different locations and combinations of heat source strengths of the baffles for a fixed Grashof number of 106and Prandtl number of 0.71. The coupled governing equations were solved bya finite-difference method using alternating direction implicit technique and successive overrelaxation methods. The obtained results clearly show that the hydrodynamic and thermal fields in the cavity depend on both the location and strength of the heat-generating baffles. Though the flow inhibition is caused by both the baffles, the baffle with higher source strength plays a decisive role in inducing the flow. The locations of baffles with unequal source strengths produce significant changes in the net heat transfer rate. This is further magnified for higher contrast in source strengths. This study provides qualitative suggestions that may improve the thermal design of sealed enclosures, which are encountered frequently in the electronics industry.

Comprehensive Numerical Analysis of Finite Difference Time Domain Methods for Improving Optical Waveguide Sensor Accuracy

This paper discusses numerical analysis methods for different geometrical features that have limited interval values for typically used sensor wavelengths. Compared with existing Finite Difference Time Domain (FDTD) methods, the alternating direction implicit (ADI)-FDTD method reduces the number of sub-steps by a factor of two to three, which represents a 33% time savings in each single run. The local one-dimensional (LOD)-FDTD method has similar numerical equation properties, which should be calculated as in the previous method. Generally, a small number of arithmetic processes, which result in a shorter simulation time, are desired. The alternating direction implicit technique can be considered a significant step forward for improving the efficiency of unconditionally stable FDTD schemes. This comparative study shows that the local one-dimensional method had minimum relative error ranges of less than 40% for analytical frequencies above 42.85 GHz, and the same accuracy was generated by both methods.

Multistep finite difference schemes for the variable coefficient delay parabolic equations

In this paper, we study the high efficient numerical methods to solve a class of variable coefficient delay parabolic differential equations. We propose two types of multistep finite difference schemes while prove the solvability, convergence and stability of both schemes. The convergence orders are and , respectively, in the sense of -norm. By some new analytical transformations, we extend our schemes to solve a more general variable coefficient delay convection–diffusion–reaction equations and also apply them to the higher dimensional cases. Furthermore, multistep finite difference method and compact multistep finite difference method for two dimensional variable coefficient diffusion–reaction equations are proposed and the suitable alternate direction implicit (ADI) technique is constructed for the multistep finite difference scheme. At last, several numerical experiments are carried out to illustrate the effectiveness of both schemes.

Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles. When the ADI technique is coupled with orthogonal spline collocation (OSC) for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order. In [2], we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms. A natural question that arises is: Does this method have an extension to non-rectangular regions? In this paper, we present a simple idea of how the ADI OSC technique can be extended to some such regions. Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem (TPBVP). We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.

Sensitivity study with respect to direction of ADI method during re-flooding in AHWR

Abstract The Advanced Heavy water Reactor (AHWR) is a natural circulation vertical pressure tube type boiling light water cooled and heavy water moderated reactor. As the AHWR fuel bundle quenching under accident condition is designed primarily with radial jets at several axial locations, bottom re-flooding still remain open as another option. Radial direction injection of emergency core cooling leads to rewetting of AHWR fuel cluster in circumferential direction. A 3D fuel pin model has been developed by using Finite Difference Method (FDM) of transient heat conduction equation. Alternating Direction Implicit technique of Finite Difference Method (FDM) has been used for discretisation of numerical equation in different time step at different direction. Sensitivity numerical study with respect to direction of ADI method has been carried out to optimize the time step during the transient as well as steady state and is found that it is insensitivity with direction of solution. Further, to assess influence of circumferential rewetting vis-à-vis axial rewetting. Both the analyses are carried out with same fluid temperature and heat transfer coefficients as boundary conditions. It has been found from the analyses that for radial jet, the circumferential conduction is significant and overall the fuel temperature falls in the quench plane with the initiation of quenching event. The paper discusses the sensitivity study with respect to direction of ADI solution and comparison of numerical results for circumferential and axial rewetting for single pin.

Internal natural convection driven by an orthogonal pair of differentially heated plates

An internal natural convective flow arising in a closed square cavity due to the presence of two orthogonal and differentially heated plates is studied numerically. The flow is assumed to be laminar and two-dimensional. The Alternating Direction Implicit technique in association with the Successive Over Relaxation method is used to solve the coupled nonlinear governing equations. The heat and fluid flow interactions are studied for various combinations of plate–plate and plate–wall temperature ratios and different positions of the heated plates. When there are two plates with high temperature contrast the heat transfer mechanism inside the cavity is mainly ruled by the hotter one. In general convection heat transfer becomes strengthened when the vertical plate is hotter.

Two-dimensional prediction of groundwater contaminants near dumpsites: a case of Soluos dumpsite in Alimosho Local Government, Lagos State, Nigeria

The lack of proper management of solid wastes in modern society poses a threat to groundwater integrity. A two-dimensional transport model was used to predict the level of a set of 12 number contaminant species near Soluos dumpsite by employing alternating direction implicit technique implemented in Matlab 7.9. The model parameters obtained through sensitivity analysis of model parameters of Yadav and Jaiswal were used in this work. The predicted results showed a regular trend of decreasing contaminant concentration as the distance increases from the dumpsite and were validated with the field data from Soluos dumpsite. It was revealed that the two-dimensional transport model predicted above 86% confidence level which is an indication that the model parameters used in this work are suitable for the Soluos dumpsite.

Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations

In this paper, we study the simulation of nonlinear Schrödinger equation in one, two and three dimensions. The proposed method is based on a time-splitting method that decomposes the original problem into two parts, a linear equation and a nonlinear equation. The linear equation in one dimension is approximated with the Chebyshev pseudo-spectral collocation method in space variable and the Crank–Nicolson method in time; while the nonlinear equation with constant coefficients can be solved exactly. As the goal of the present paper is to study the nonlinear Schrödinger equation in the large finite domain, we propose a domain decomposition method. In comparison with the single-domain, the multi-domain methods can produce a sparse differentiation matrix with fewer memory space and less computations. In this study, we choose an overlapping multi-domain scheme. By applying the alternating direction implicit technique, we extend this efficient method to solve the nonlinear Schrödinger equation both in two and three dimensions, while for the solution at each time step, it only needs to solve a sequence of linear partial differential equations in one dimension, respectively. Several examples for one- and multi-dimensional nonlinear Schrödinger equations are presented to demonstrate high accuracy and capability of the proposed method. Some numerical experiments are reported which show that this scheme preserves the conservation laws of charge and energy.

Suppression and reversal of drop formation on horizontal cylinders due to surfactant convection

When a thin liquid film is applied to the surface of a horizontal cylinder, gravity will cause a drainage of liquid from the top and sides of the cylinder towards the cylinder bottom. If surfactant is present on the surface of the film, this will cause a convection of surfactant resulting in a higher concentration of surfactant on the cylinder bottom compared to the top and sides of the cylinder. The result is a surface tension gradient, which is equivalent to a surface shear stress, and acts to oppose the drainage of the coating layer due to gravity. For sufficiently small cylinders, this cannot only slow the drainage but reverse the flow, causing a net flux of liquid upward from the bottom of the cylinder towards the top of the cylinder. If this flux is sufficiently strong, a “collar” of liquid forms around the cylinder. In this paper, we develop a mathematical model, based on the lubrication approximations, of the gravitational, surface tension, and surface tension gradient forces, and their effects on the evolution of a thin liquid film coating a horizontal circular cylinder. Using finite differences and an alternating direction implicit technique, numerical simulations show that even for comparatively weak surfactants, surface tension gradient effects greatly affect the flow history and must be included to accurately model the evolution of the film. They cannot only slow the drainage of liquid towards a pendant drop on the bottom of the cylinder, but reverse the flux, resulting in a thicker coating on the top of the cylinder compared to the surfactant-free case. Results from the simulation are presented over a wide range of the dimensionless parameters which characterize the problem.

Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

AbstractThe alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles. When the ADI technique is coupled with orthogonal spline collocation (OSC) for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order. In [2], we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms. A natural question that arises is: Does this method have an extension to non-rectangular regions? In this paper, we present a simple idea of how the ADI OSC technique can be extended to some such regions. Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem (TPBVP). We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.

Optimal design of an in-situ bioremediation system using support vector machine and particle swarm optimization

A methodology based on support vector machine and particle swarm optimization techniques (SVM-PSO) was used in this study to determine an optimal pumping rate and well location to achieve an optimal cost of an in-situ bioremediation system. In the first stage of the two stage methodology suggested for optimal in-situ bioremediation design, the optimal number of wells and their locations was determined from preselected candidate well locations. The pumping rate and well location in the first stage were subsequently optimized in the second stage of the methodology. The highly nonlinear system of equations governing in-situ bioremediation comprises the equations of flow and solute transport coupled with relevant biodegradation kinetics. A finite difference model was developed to simulate the process of in-situ bioremediation using an Alternate-Direction Implicit technique. This developed model (BIOFDM) yields the spatial and temporal distribution of contaminant concentration for predefined initial and boundary conditions. BIOFDM was later validated by comparing the simulated results with those obtained using BIOPLUME III for the case study of Shieh and Peralta (2005). The results were found to be in close agreement. Moreover, since the solution of the highly nonlinear equation otherwise requires significant computational effort, the computational burden in this study was managed within a practical time frame by replacing the BIOFDM model with a trained SVM model. Support Vector Machine which generates fast solutions in real time was considered to be a universal function approximator in the study. Apart from reducing the computational burden, this technique generates a set of near optimal solutions (instead of a single optimal solution) and creates a re-usable data base that could be used to address many other management problems. Besides this, the search for an optimal pumping pattern was directed by a simple PSO technique and a penalty parameter approach was adopted to handle the constraints in the PSO. The results showed that the costs involved in the proposed management solution were consistent with that resulting from other nontraditional optimization techniques which use embedded/linked bioremediation simulation models. Moreover, an optimal transient pumping strategy resulted in an overall reduction in pumping cost by almost 20% when compared to cases where a steady state pumping strategy was adopted. A considerable reduction in the number of simulations was achieved using the SVM approach.

Spherical ADI FDTD Method With Application to Propagation in the Earth Ionosphere Cavity

The alternating direction implicit (ADI) finite-difference time-domain (FDTD) technique is implemented using the spherical coordinate system and applied to wave propagation in the lossy Earth ionosphere cavity. The paper shows, for the first time, that the ADI technique can be developed in conjunction with spherical FDTD in order to overcome the typically long computing times involved in Earth ionosphere cavity problems. Numerical accuracy of the full wave three-dimensional (3D) ADI implementation is validated by comparison with spherical FDTD results. While not free from late time instability, transient responses are shown to be obtained accurately for time steps as large as 15 times the time step derived from cells at the Equator. Hence the spherical ADI is demonstrated to be a promising technique to efficiently characterize Schumann resonances and other complex propagation phenomena around the Earth, while overcoming the space cell eccentricity typically generated by spherical grids.